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Michael Kinyon

Publications and source records attributed to Michael Kinyon.

At least 55 records · Page 3Linked to original sources
arXiv2012

Torsors and ternary Moufang loops arising in projective geometry

We give an interpretation of the construction of torsors from preceding work (Bertram, Kinyon: Associative Geometries. I, J. Lie Theory 20) in terms of classical projective geometry. For the Desarguesian case, this leads to a reformulation of certain results from lot.cit., whereas for the Moufang case the result is new. But even in the Desarguesian case it sheds new light on the relation between the lattice structure and the algebraic structures of a projective space.

math.GR↗
arXiv2012

Pseudoautomorphisms of Bruck loops and their generalizations

We show that in a weak commutative inverse property loop, such as a Bruck loop, if $α$ is a right [left] pseudoautomorphism with companion $c$, then $c$ [$c^2$] must lie in the left nucleus. In particular, for any such loop with trivial left nucleus, every right pseudoautomorphism is an automorphism and if the squaring map is a permutation, then every left pseudoautomorphism is an automorphism as well. We also show that every pseudoautomorphism of a commutative inverse property loop is an automorphism, generalizing a well-known result of Bruck.

math.GR↗
arXiv2011

Incidence properties of cosets in loops

We study incidence properties among cosets of finite loops, with emphasis on well-structured varieties such as antiautomorphic loops and Bol loops. While cosets in groups are either disjoint or identical, we find that the incidence structure in general loops can be much richer. Every symmetric design, for example, can be realized as a canonical collection of cosets of a finite loop. We show that in the variety of antiautomorphic loops the poset formed by set inclusion among intersections of left cosets is isomorphic to that formed by right cosets. We present an algorithm that, given a finite Bol loop $S$, can in some cases determine whether $|S|$ divides $|Q|$ for all finite Bol loops $Q$ with $S \le Q$, and even whether there is a selection of left cosets of $S$ that partitions $Q$. This method results in a positive confirmation of Lagrange's Theorem for Bol loops for a few new cases of subloops. Finally, we show that in a left automorphic Moufang loop $Q$ (in particular, in a commutative Moufang loop $Q$), two left cosets of $S\le Q$ are either disjoint or they intersect in a set whose cardinality equals that of some subloop of $S$.

math.CO↗
arXiv2011

A characterization of adequate semigroups by forbidden subsemigroups

A semigroup is \emph{amiable} if there is exactly one idempotent in each $\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. A semigroup is \emph{adequate} if it is amiable and if its idempotents commute. We characterize adequate semigroups by showing that they are precisely those amiable semigroups which do not contain isomorphic copies of two particular nonadequate semigroups as subsemigroups.

math.GR↗
arXiv2011

When is a Bol loop Moufang?

There are a number of identities which, if satisfied by a Bol loop, imply that the loop is actually Moufang. In this paper we show that in a number of cases, the Moufang identity is also forced not by a single identity, but by giving elements a choice of equations to satisfy.

math.GR↗
arXiv2011

Uniquely 2-divisible Bol loops

Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.

math.GR↗
arXiv2011

On a problem of M. Kambites regarding abundant semigroups

A semigroup is \emph{regular} if it contains at least one idempotent in each $\mathcal{R}$-class and in each $\mathcal{L}$-class. A regular semigroup is \emph{inverse} if satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each $\mathcal{R}$-class and in each $\mathcal{L}$-class, or (ii) the idempotents commute. Analogously, a semigroup is \emph{abundant} if it contains at least one idempotent in each $\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. An abundant semigroup is \emph{adequate} if its idempotents commute. In adequate semigroups, there is a unique idempotent in each $\mathcal{R}^*$ and $\mathcal{L}^*$-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each $\mathcal{R}^*$ and $\mathcal{L}^*$-class, must the idempotents commute? In this note we use ideal extensions to provide a negative answer to this question.

math.GR↗
arXiv2011

Independent axiom systems for nearlattices

A nearlattice is a join semilattice such that every principal filter is a lattice with respect to the induced order. Hickman and later Chajda et al independently showed that nearlattices can be treated as varieties of algebras with a ternary operation satisfying certain axioms. Our main result is that the variety of nearlattices is 2-based, and we exhibit an explicit axiom system of two independent identities. We also show that the original axiom systems of Hickman and of Chajda et al are, respectively, dependent.

math.CO↗
arXiv2011

Nilpotency in automorphic loops of prime power order

A loop is automorphic if its inner mappings are automorphisms. Using so-called associated operations, we show that every commutative automorphic loop of odd prime power order is centrally nilpotent. Starting with anisotropic planes in the vector space of $2\times 2$ matrices over the field of prime order $p$, we construct a family of automorphic loops of order $p^3$ with trivial center.

math.GR↗
arXiv2011

Ken Kunen: Algebraist

This paper is a greatly expanded version of a talk I gave in April 2009 at KunenFest. It describes Ken's work in algebra, particularly using automated deduction tools.

math.HO↗
arXiv2011

Searching for small simple automorphic loops

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no nonassociative simple commutative automorphic loop of order less than $2^{12}$, and no nonassociative simple automorphic loop of order less that 2500. We obtain examples of nonassociative simple right automorphic loops.

math.GR↗
arXiv2010

Minimal paths in the commuting graphs of semigroups

Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted $\cg(S)$, is the graph whose vertices are the non-central elements of $S$ and whose edges are the sets $\{a,b\}$ of vertices such that $a\ne b$ and $ab=ba$. Denote by $T(X)$ the semigroup of full transformations on a finite set $X$. Let $J$ be any ideal of $T(X)$ such that $J$ is different from the ideal of constant transformations on $X$. We prove that if $|X|\geq4$, then, with a few exceptions, the diameter of $\cg(J)$ is 5. On the other hand, we prove that for every positive integer $n$, there exists a semigroup $S$ such that the diameter of $\cg(S)$ is $n$. We also study the left paths in $\cg(S)$, that is, paths $a_1-a_2-...-a_m$ such that $a_1\ne a_m$ and $a_1a_i=a_ma_i$ for all $i\in \{1,\ldot, m\}$. We prove that for every positive integer $n\geq2$, except $n=3$, there exists a semigroup whose shortest left path has length $n$. As a corollary, we use the previous results to solve a purely algebraic old problem posed by B.M. Schein.

math.GR↗
arXiv2010

An elegant 3-basis for inverse semigroups

It is well known that in every inverse semigroup the binary operation and the unary operation of inversion satisfy the following three identities: [\quad x=(xx')x \qquad \quad (xx')(y'y)=(y'y)(xx') \qquad \quad (xy)z=x(yz"). ] The goal of this note is to prove the converse, that is, we prove that an algebra of type $<2,1>$ satisfying these three identities is an inverse semigroup and the unary operation coincides with the usual inversion on such semigroups.

math.GR↗
arXiv2010

Associative Geometries. I: Torsors, linear relations and Grassmannians

We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized projective geometries, where the former correspond to the Lie product of an associative algebra and the latter to its Jordan product. A further development of the theory encompassing involutive associative algebras will be given in subsequent work.

math.RA↗
arXiv2010

Associative Geometries. II: Involutions, the classical torsors, and their homotopes

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative geometries. We prove that, under suitable assumptions, the groups and their homotopes have a canonical semigroup completion.

math.RA↗
arXiv2010

Associative Geometries. I: Torsors, linear relations and Grassmannians

We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized projective geometries, where the former correspond to the Lie product of an associative algebra and the latter to its Jordan product. A further development of the theory encompassing involutive associative algebras will be given in subsequent work.

math.RA↗
arXiv2010

Associative Geometries. II: Involutions, the classical torsors, and their homotopes

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative geometries. We prove that, under suitable assumptions, the groups and their homotopes have a canonical semigroup completion.

math.RA↗